Solve The Inequality $-9.4 \geq 1.7x + 4.2$. What Is The Solution For $x$?

Introduction

In this article, we will walk through the step-by-step process of solving the linear inequality $-9.4 \geq 1.7x + 4.2$. Understanding how to solve inequalities is a fundamental skill in mathematics, with applications ranging from basic algebra to more advanced calculus and real-world problem-solving. Inequalities are mathematical statements that compare two expressions using symbols such as $\geq$ (greater than or equal to), $\leq$ (less than or equal to), $>$ (greater than), and $<$ (less than). Unlike equations, which have a single solution or a set of solutions, inequalities typically have a range of values that satisfy the statement. This makes them incredibly useful for modeling situations where constraints and boundaries are important, such as in optimization problems and constraint satisfaction.

Our primary goal is to isolate the variable $x$ on one side of the inequality to determine the range of values that make the inequality true. We'll achieve this by applying algebraic operations while adhering to the rules that govern inequalities. These rules include maintaining the direction of the inequality when adding or subtracting values from both sides and flipping the inequality sign when multiplying or dividing both sides by a negative number. This comprehensive guide aims to provide a clear and detailed explanation of each step involved in solving the given inequality, making it accessible to learners of all levels. We will also discuss how to interpret the solution and represent it graphically, further solidifying understanding. This methodical approach will not only help in solving the current problem but also equip you with the skills to tackle similar inequality problems with confidence.

Step-by-Step Solution

1. Isolate the Term with $x$

To begin solving the inequality $-9.4 \geq 1.7x + 4.2$, our first objective is to isolate the term containing the variable $x$. In this case, the term is $1.7x$. To achieve this, we need to eliminate the constant term on the right side of the inequality, which is $+4.2$. The method to eliminate this term is to perform the inverse operation. Since $4.2$ is being added, we subtract $4.2$ from both sides of the inequality. This ensures that the inequality remains balanced, as whatever operation we perform on one side, we must also perform on the other side to maintain the relationship.

Subtracting $4.2$ from both sides gives us:

$$-9.4 - 4.2 \geq 1.7x + 4.2 - 4.2$$

Performing the subtraction, we get:

$$-13.6 \geq 1.7x$$

This step is crucial because it simplifies the inequality, bringing us closer to isolating $x$. By subtracting $4.2$ from both sides, we have effectively moved the constant term to the left side, leaving the term with $x$ by itself on the right side. This prepares us for the next step, where we will isolate $x$ completely. Remember, the key to solving inequalities is to perform operations in a way that maintains the balance and the truth of the inequality. By systematically eliminating terms, we can eventually isolate the variable and find the solution set.

2. Isolate $x$

After subtracting $4.2$ from both sides, we arrived at the inequality $-13.6 \geq 1.7x$. Our next goal is to isolate $x$ completely. Currently, $x$ is being multiplied by $1.7$. To undo this multiplication, we need to perform the inverse operation, which is division. We will divide both sides of the inequality by $1.7$ to isolate $x$. It's crucial to remember a key rule when working with inequalities: if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. However, in this case, we are dividing by a positive number ($1.7$), so we do not need to reverse the inequality sign.

Dividing both sides by $1.7$ gives us:

$$\frac{-13.6}{1.7} \geq \frac{1.7x}{1.7}$$

Now, we perform the division:

$$-8 \geq x$$

This simplifies to:

$$x \leq -8$$

This result tells us that $x$ is less than or equal to $-8$. In other words, any value of $x$ that is $-8$ or smaller will satisfy the original inequality. This step is pivotal in solving the inequality, as it directly reveals the range of possible values for $x$. The division by $1.7$ effectively isolates $x$, providing a clear solution to the inequality. Always double-check whether you need to reverse the inequality sign when dividing or multiplying by a negative number to ensure the solution is accurate.

Final Answer

Therefore, the solution to the inequality $-9.4 \geq 1.7x + 4.2$ is $x \leq -8$. This matches option A.

Conclusion

In this detailed walkthrough, we have successfully solved the inequality $-9.4 \geq 1.7x + 4.2$. By systematically isolating the variable $x$, we found that the solution is $x \leq -8$. This process involved subtracting a constant from both sides and then dividing by the coefficient of $x$, adhering to the rules that govern inequalities. Specifically, we maintained the direction of the inequality sign because we divided by a positive number.

Understanding how to solve inequalities is a crucial skill in mathematics. Inequalities are used extensively in various fields, including engineering, economics, and computer science, to model constraints and determine feasible regions. For example, in optimization problems, inequalities often define the boundaries within which a solution must lie.

The steps we followed in this solution are applicable to a wide range of linear inequalities. The key is to perform inverse operations to isolate the variable while maintaining the balance of the inequality. Always remember to reverse the inequality sign when multiplying or dividing by a negative number. The solution $x \leq -8$ means that any value of $x$ that is less than or equal to $-8$ will satisfy the original inequality. This can be visualized on a number line, where the solution is represented by a closed circle at $-8$ and an arrow extending to the left, indicating all values less than $-8$.

By mastering the techniques presented here, you can confidently approach and solve a variety of inequality problems. The ability to solve inequalities not only enhances mathematical proficiency but also provides a valuable tool for problem-solving in numerous real-world scenarios. Practice and careful attention to detail are key to success in this area of mathematics.